Interval Estimation and Hypothesis Testing in Statistics
3.1 Interval Estimation
3.2 Hypothesis Tests
3.3 Rejection Regions for Specific Alternatives
3.4 Examples of Hypothesis Tests
The two endpoints provide an interval estimator.
In repeated sampling 95% of the intervals constructed this way will contain the
true value of the parameter
β
2
.
This easy derivation of an interval estimator is based on the assumption SR6
and
that we know the variance of the error term
σ
2
.
Replacing
σ
2
with creates a random variable
t:
The ratio has a
t
-distribution with (
N –
2) degrees of freedom,
which we denote as .
In general we can say, if assumptions SR1-SR6 hold in the simple linear regression
model, then
The
t
-distribution is a bell shaped curve centered at zero.
It looks like the standard normal distribution, except it is more spread out, with a
larger variance and thicker tails.
The shape of the
t
-distribution is controlled by a single parameter called the
degrees
of freedom
, often abbreviated as
df
.
We can find a “critical value” from a
t-
distribution such that
where α is a probability often taken to be α = .01 or α = .05.
The critical value
t
c
for degrees of freedom
m
is the percentile value .
Figure 3.1 Critical Values from a
t
-distribution
Each shaded “tail” area contains
/2 of the probability, so that 1–
α
of the
probability is contained in the center portion.
Consequently, we can make the probability statement
For the food expenditure data
The critical value
t
c
= 2.024, which is appropriate for
= .05 and 38 degrees of
freedom.
To construct an interval estimate for
2
we use the least squares estimate
b
2
= 10.21
and its standard error
A “95% confidence interval estimate” for
2
:
When the procedure we used is applied to many random samples of data from the
same population, then 95% of all the interval estimates constructed using this
procedure will contain the true parameter.
Components of Hypothesis Tests
1.
A null hypothesis,
H
0
2.
An alternative hypothesis,
H
1
3.
A test statistic
4.
A rejection region
5.
A conclusion
The Null Hypothesis
The null hypothesis, which is denoted
H
0
(
H-naught
), specifies a value for a
regression parameter.
The null hypothesis is stated , where
c
is a constant, and is an important
value in the context of a specific regression model.
The Alternative Hypothesis
Paired with every null hypothesis is a logical alternative hypothesis,
H
1
, that we
will accept if the null hypothesis is rejected.
For the null hypothesis
H
0
:
k
=
c
the three possible alternative hypotheses are:
The Test Statistic
If
the null hypothesis is
true
,
then
we can substitute
c
for
k
and it
follows that
If the null hypothesis is
not true
,
then the
t-
statistic in (3.7) does
not
have a
t
-
distribution with
N
2 degrees of freedom.
The Rejection Region
The rejection region depends on the form of the alternative. It is the range of values
of the test statistic that leads to
rejection
of the null hypothesis. It is possible to
construct a rejection region only if we have:
a test statistic whose distribution is known when the null hypothesis is true
an alternative hypothesis
a level of significance
The level of significance
α
is usually chosen to be .01, .05 or .10.
A Conclusion
We make a correct decision if:
The null hypothesis is
false
and we decide to
reject
it.
The null hypothesis is
true
and we decide
not
to reject it.
Our decision is incorrect if:
The null hypothesis is
true
and we decide to
reject
it (a Type I error)
The null hypothesis is
false
and we decide
not
to reject it (a Type II error)
3.3.1. One-tail Tests with Alternative “Greater Than” (>)
3.3.2. One-tail Tests with Alternative “Less Than” (<)
3.3.3. Two-tail Tests with Alternative “Not Equal To” (≠)
Figure
3.2
Rejection region for a one-tail test of
H
0
:
β
k
=
c
against
H
1
:
β
k
>
c
Figure
3.3
The rejection region for a one-tail test of
H
0
:
β
k
=
c
against
H
1
:
β
k
<
c
Figure
3.4
The rejection region for a two-tail test of
H
0
:
β
k
=
c
against
H
1
:
β
k
≠
c
The concept of interval estimation and hypothesis testing in statistics involves techniques such as constructing interval estimators, performing hypothesis tests, determining critical values from t-distributions, and making probability statements. Assumptions must be met in linear regression models for accurate estimation. The t-distribution, with degrees of freedom influencing its shape, plays a crucial role in statistical inference.
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3.1 Interval Estimation 3.2 Hypothesis Tests 3.3 Rejection Regions for Specific Alternatives 3.4 Examples of Hypothesis Tests
Assumptions of the Simple Linear Regression Model = + + y E e = var( ) cov( , x e SR1. SR2. SR3. SR4. SR5. The variable x is not random, and must take at least two different values. SR6. (optional) The values of e are normally distributed about their mean 1 2 = + y ) y y i j ( ) 0 = = ) e e i j ( ) var( ) cov( , E y x 1 2 2 e = = 0 2 ~ ( , 0 ) e N
( ) The two endpoints provide an interval estimator. ( 2 1.96 b x ) 2 2 x i In repeated sampling 95% of the intervals constructed this way will contain the true value of the parameter 2. This easy derivation of an interval estimator is based on the assumption SR6 and that we know the variance of the error term 2.
2 Replacing 2with creates a random variable t: b b b = = = 2 2 2 2 2 2 ~ t t ( ) b (3.2) ( 2) N se ( ) ( ) b 2 2 x x var 2 i 2 The ratio has a t-distribution with (N 2) degrees of freedom, ( ) ( ) 2 2 2 se t b b = ~ N t t which we denote as . ( 2)
In general we can say, if assumptions SR1-SR6 hold in the simple linear regression model, then b = = k k ~ for 1,2 t t k ( ) b (3.3) ( ) 2 N se k The t-distribution is a bell shaped curve centered at zero. It looks like the standard normal distribution, except it is more spread out, with a larger variance and thicker tails. The shape of the t-distribution is controlled by a single parameter called the degrees of freedom, often abbreviated as df.
We can find a critical value from a t-distribution such that ( ) ( ) = = 2 P t t P t t c c where is a probability often taken to be = .01 or = .05. t The critical value tcfor degrees of freedom m is the percentile value . ( ) 1 2,m
Each shaded tail area contains /2 of the probability, so that 1 of the probability is contained in the center portion. Consequently, we can make the probability statement ) 1 = ( P t t t (3.4) c c b ] 1 = k k [ P t t c c se( ) b k + = [ se( ) b se( )] 1 k b P b t b t (3.5) k c k k k c
For the food expenditure data + = [ 2.024se( ) 2.024se( )] .95 b P b b b (3.6) 2 2 2 2 2 The critical value tc= 2.024, which is appropriate for = .05 and 38 degrees of freedom. To construct an interval estimate for 2we use the least squares estimate b2= 10.21 and its standard error = = = se( ) b var( ) 4.38 2.09 b 2 2
A 95% confidence interval estimate for 2: se( ) 10.21 2.024(2.09)=[5.97,14.45] b = b t 2 2 c When the procedure we used is applied to many random samples of data from the same population, then 95% of all the interval estimates constructed using this procedure will contain the true parameter.
Components of Hypothesis Tests A null hypothesis, H0 An alternative hypothesis, H1 1. 2. 3. A test statistic A rejection region 4. A conclusion 5.
The Null Hypothesis The null hypothesis, which is denoted H0(H-naught), specifies a value for a regression parameter. The null hypothesis is stated , where c is a constant, and is an important k H c = 0: value in the context of a specific regression model.
The Alternative Hypothesis Paired with every null hypothesis is a logical alternative hypothesis, H1, that we will accept if the null hypothesis is rejected. For the null hypothesis H0: k= c the three possible alternative hypotheses are: 1: H c k 1: H c k 1: H c k
The Test Statistic ( k t b = ) se( ) ~ b t ( 2) k k N = If the null hypothesis is true, then we can substitute c for kand it follows that 0: H c k b c = k ~ t t (3.7) ( 2) N se( ) b k If the null hypothesis is not true, then the t-statistic in (3.7) does not have a t- distribution with N 2 degrees of freedom.
The Rejection Region The rejection region depends on the form of the alternative. It is the range of values of the test statistic that leads to rejection of the null hypothesis. It is possible to construct a rejection region only if we have: a test statistic whose distribution is known when the null hypothesis is true an alternative hypothesis a level of significance The level of significance is usually chosen to be .01, .05 or .10.
A Conclusion We make a correct decision if: The null hypothesis is false and we decide to reject it. The null hypothesis is true and we decide not to reject it. Our decision is incorrect if: The null hypothesis is true and we decide to reject it (a Type I error) The null hypothesis is false and we decide not to reject it (a Type II error)
3.3.1. One-tail Tests with Alternative Greater Than (>) 3.3.2. One-tail Tests with Alternative Less Than (<) 3.3.3. Two-tail Tests with Alternative Not Equal To ( )
Figure 3.2 Rejection region for a one-tail test of H0: k= c against H1: k> c
= When testing the null hypothesis against the 0: H c k alternative hypothesis , reject the null hypothesis k H 1: c and accept the alternative hypothesis if . t t ( ) 1 , 2 N
Figure 3.3 The rejection region for a one-tail test of H0: k= c against H1: k< c
= When testing the null hypothesis against the 0: H c k alternative hypothesis , reject the null hypothesis k H 1: c and accept the alternative hypothesis if . t t ( ) , 2 N
Figure 3.4 The rejection region for a two-tail test of H0: k= c against H1: k c
= When testing the null hypothesis against the 0: H c k alternative hypothesis , reject the null hypothesis k H 1: c and accept the alternative hypothesis if or if t t ( ) 2, 2 N . t t ( ) 1 2, 2 N
STEP-BY-STEP PROCEDURE FOR TESTING HYPOTHESES 1. Determine the null and alternative hypotheses. 2. Specify the test statistic and its distribution if the null hypothesis is true. 3. Select and determine the rejection region. 4. Calculate the sample value of the test statistic. 5. State your conclusion.
